SOLUTION: For certain values of k and m, the system 3a + 2b = 2 6a + 2b = k - 3a - mb has infinitely many solutions (a,b). What are k and m?

Question 1204913: For certain values of k and m, the system
3a + 2b = 2
6a + 2b = k - 3a - mb
has infinitely many solutions (a,b). What are k and m?
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Answers: k = 6 and m = 4

Explanation

Add 3a to both sides and add mb to both sides to get the 'a's and 'b's together
6a + 2b = k - 3a - mb
6a + 3a + 2b + mb = k
9a + (2 + m)b = k

This system

is equivalent to this system

The jump from 3a to 9a is "times 3", so let's triple everything in the 1st equation
3a + 2b = 2 triples to 9a + 6b = 6

We now have this equivalent system

To have the coefficients for b to match up, we must have be the case. That solves to
We must have so the right hand sides match up.

In short: k = 6 and m = 4
Let's plug those into the original 2nd equation
6a + 2b = k - 3a - m*b
6a + 2b = 6 - 3a - 4b
6a + 2b + 3a + 4b = 6
9a + 6b = 6
3(3a + 2b) = 6
3a + 2b = 6/3
3a + 2b = 2
We arrive at the 1st original equation.
It proves that equation2 is the same as equation1 when k = 6 and m = 4.
The answers are confirmed.

Another way to confirm the answers is to graph the equations 3x+2y = 2 and 6x+2y = k - 3x - m*y
I have replaced 'a' with x, and b with y.
After inputting those equations into Desmos or GeoGebra, set the parameters k = 6 and m = 4 to notice one line overlaps the other perfectly.
Link to interactive Desmos graph
https://www.desmos.com/calculator/rzgnjxxyfc
Click the button for one of the equations to turn the line on/off. Do this repeatedly. The line will blink different colors to show the overlap.

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

[1]

[2]

Equation [1] is equivalent to

[3]

There are an infinite number of solutions if [2] and [3] are equivalent. That is true only if the coefficients of b in both equations are the same and the constants in both are the same.

2+m = 6 --> m = 4
k = 6

ANSWERS: k = 6; m = 4

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